We study bounded derived categories of the category of representations of infinite quivers over a ring $R$. In case $R$ is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left, resp. right, rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

Combinatorial structures that compose and decompose give rise to Hopf monoids
in Joyal's category of species. The Hadamard product of two Hopf monoids
is another Hopf monoid. We prove two main results regarding freeness of
Hadamard products. The first one states
that if one factor is connected and the other is free as a monoid,
their Hadamard product is free (and connected).
The second provides an explicit basis for the Hadamard
product when both factors are free.
The first main result is obtained by showing the existence of a one-parameter deformation
of the comonoid structure and appealing to a rigidity result of Loday and Ronco
that applies when the parameter is set to zero.
To obtain the second result, we introduce an operation on species that is intertwined
by the free monoid functor with the Hadamard product.
As an application of the first result, we deduce that the Boolean transform
of the dimension sequence of a connected Hopf monoid is nonnegative.

"Co-Frobenius" coalgebras were introduced as dualizations of
Frobenius algebras.
We previously showed
that they admit
left-right symmetric characterizations analogue to those of Frobenius
algebras. We consider the more general quasi-co-Frobenius (QcF)
coalgebras; the first main result in this paper is that these also
admit symmetric characterizations: a coalgebra is QcF if it is weakly
isomorphic to its (left, or right) rational dual $Rat(C^*)$, in the
sense that certain coproduct or product powers of these objects are
isomorphic. Fundamental results of Hopf algebras, such as the
equivalent characterizations of Hopf algebras with nonzero integrals
as left (or right) co-Frobenius, QcF, semiperfect or with nonzero
rational dual, as well as the uniqueness of integrals and a short
proof of the bijectivity of the antipode for such Hopf algebras all
follow as a consequence of these results. This gives a purely
representation theoretic approach to many of the basic fundamental
results in the theory of Hopf algebras. Furthermore, we introduce a
general concept of Frobenius algebra, which makes sense for infinite
dimensional and for topological algebras, and specializes to the
classical notion in the finite case. This will be a topological
algebra $A$ that is isomorphic to its complete topological dual
$A^\vee$. We show that $A$ is a (quasi)Frobenius algebra if and only
if $A$ is the dual $C^*$ of a (quasi)co-Frobenius coalgebra $C$. We
give many examples of co-Frobenius coalgebras and Hopf algebras
connected to category theory, homological algebra and the newer
q-homological algebra, topology or graph theory, showing the
importance of the concept.

Following Radford's proof of Lagrange's theorem for pointed Hopf algebras,
we prove Lagrange's theorem for Hopf monoids in the category of
connected species.
As a corollary, we obtain necessary conditions for a given subspecies
$\mathbf k$ of a Hopf monoid $\mathbf h$ to be a Hopf submonoid: the quotient of
any one of the generating series of $\mathbf h$ by the corresponding
generating series of $\mathbf k$ must have nonnegative coefficients. Other
corollaries include a necessary condition for a sequence of
nonnegative integers to be the
dimension sequence of a Hopf monoid
in the form of certain polynomial inequalities, and of
a set-theoretic Hopf monoid in the form of certain linear inequalities.
The latter express that the binomial transform of the sequence must be nonnegative.

This paper gives a characterization of homotopy fibres of inverse
image maps on groupoids of torsors that are induced by geometric
morphisms, in terms of both pointed torsors and pointed cocycles,
suitably defined. Cocycle techniques are used to give a complete
description of such fibres, when the underlying geometric morphism is
the canonical stalk on the classifying topos of a profinite group
$G$. If the torsors in question are defined with respect to a constant
group $H$, then the path components of the fibre can be identified with
the set of continuous maps from the profinite group $G$ to the group
$H$. More generally, when $H$ is not constant, this set of path components
is the set of continuous maps from a pro-object in sheaves of
groupoids to $H$, which pro-object can be viewed as a ``Grothendieck
fundamental groupoid".

We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: K-theory and Bredon cohomology for certain coefficient diagrams.

In this article we study injective representations of infinite
quivers. We classify the indecomposable injective representations of
trees and describe Gorenstein injective and projective
representations of barren trees.

We define a family of formal Khovanov brackets
of a colored link depending on two parameters.
The isomorphism classes of these brackets are
invariants of framed colored links.
The Bar-Natan functors applied to these brackets
produce Khovanov and Lee homology theories categorifying the colored
Jones polynomial. Further,
we study conditions under which
framed colored link cobordisms induce chain transformations between
our formal brackets. We conjecture that
for special choice of parameters, Khovanov and Lee homology theories
of colored links are functorial (up to sign).
Finally, we extend the Rasmussen invariant to links and give examples
where this invariant is a stronger obstruction to sliceness
than the multivariable Levine--Tristram signature.

We prove that for a topological operad $P$ the operad of oriented
cubical singular chains, $C^{\ord}_\ast(P)$, and the operad of
simplicial singular chains, $S_\ast(P)$, are weakly equivalent. As
a consequence, $C^{\ord}_\ast(P\nsemi\mathbb{Q})$ is formal if and only
if $S_\ast(P\nsemi\mathbb{Q})$ is formal, thus linking together some
formality results which are spread out in the literature. The proof
is based on an acyclic models theorem for monoidal functors. We
give different variants of the acyclic models theorem and apply
the contravariant case to study the cohomology theories for
simplicial sets defined by $R$-simplicial differential graded
algebras.

In previous papers, Barr and Raphael investigated the situation of a
topological space $Y$ and a subspace $X$ such that the induced map
$C(Y)\to C(X)$ is an epimorphism in the category $\CR$ of commutative
rings (with units). We call such an embedding a $\CR$-epic embedding
and we say that $X$ is absolute $\CR$-epic if every embedding of $X$
is $\CR$-epic. We continue this investigation. Our most notable
result shows that a Lindel\"of space $X$ is absolute $\CR$-epic if a
countable intersection of $\beta X$-neighbourhoods of $X$ is a $\beta
X$-neighbourhood of $X$. This condition is stable under countable
sums, the formation of closed subspaces, cozero-subspaces, and being
the domain or codomain of a perfect map. A strengthening of the
Lindel\"of property leads to a new class with the same closure
properties that is also closed under finite products. Moreover, all
\s-compact spaces and all Lindel\"of $P$-spaces satisfy this stronger
condition. We get some results in the non-Lindel\"of case that are
sufficient to show that the Dieudonn\'e plank and some closely related
spaces are absolute $\CR$-epic.

We study Tychonoff spaces $X$ with the property that, for all
topological embeddings $X\to Y $, the induced map $C(Y) \to C(X)$ is an
epimorphism of rings. Such spaces are called \good. The simplest
examples of \good spaces are $\sigma$-compact locally compact spaces and
\Lin $P$-spaces. We show that \good first countable spaces must be
locally compact.
However, a ``bad'' class of \good spaces is exhibited whose pathology
settles, in the negative, a number of open questions. Spaces which are
not \good abound, and some are presented.

We develop an explicit skein-theoretical algorithm to compute the
Alexander polynomial of a 3-manifold from a surgery presentation
employing the methods used in the construction of quantum invariants
of 3-manifolds. As a prerequisite we establish and prove a rather
unexpected equivalence between the topological quantum field theory
constructed by Frohman and Nicas using the homology of
$U(1)$-representation varieties on the one side and the
combinatorially constructed Hennings TQFT based on the quasitriangular
Hopf algebra $\mathcal{N} = \mathbb{Z}/2 \ltimes \bigwedge^*
\mathbb{R}^2$ on the other side. We find that both TQFT's are $\SL
(2,\mathbb{R})$-equivariant functors and, as such, are isomorphic.
The $\SL (2,\mathbb{R})$-action in the Hennings construction comes
from the natural action on $\mathcal{N}$ and in the case of the
Frohman--Nicas theory from the Hard--Lefschetz decomposition of the
$U(1)$-moduli spaces given that they are naturally K\"ahler. The
irreducible components of this TQFT, corresponding to simple
representations of $\SL(2,\mathbb{Z})$ and $\Sp(2g,\mathbb{Z})$, thus
yield a large family of homological TQFT's by taking sums and products.
We give several examples of TQFT's and invariants that appear to fit
into this family, such as Milnor and Reidemeister Torsion,
Seiberg--Witten theories, Casson type theories for homology circles
{\it \`a la} Donaldson, higher rank gauge theories following Frohman
and Nicas, and the $\mathbb{Z}/p\mathbb{Z}$ reductions of
Reshetikhin--Turaev theories over the cyclotomic integers $\mathbb{Z}
[\zeta_p]$. We also conjecture that the Hennings TQFT for
quantum-$\mathfrak{sl}_2$ is the product of the Reshetikhin--Turaev
TQFT and such a homological TQFT.

In this paper, we investigate projectivity in the category of operator
spaces. In particular, we show that the Fourier algebra of a locally
compact group $G$ is operator biprojective if and only if $G$ is
discrete.

In this article we state and prove precise theorems on the homotopy
classification of graded categorical groups and their homomorphisms.
The results use equivariant group cohomology, and they are applied to
show a treatment of the general equivariant group extension problem.

For every integer $n>1$ and infinite field $F$ we construct a spectral
sequence converging to the homology of $\GL_n(F)$ relative to the
group of monomial matrices $\GM_n(F)$. Some entries in $E^2$-terms of
these spectral sequences may be interpreted as a natural
generalization of the Bloch group to higher dimensions. These groups
may be characterized as homology of $\GL_n$ relatively to $\GL_{n-1}$
and $\GM_n$. We apply the machinery developed to the investigation of
stabilization maps in homology of General Linear Groups.

In this paper we show that for a Grothendieck category $\A$ and a
complex $E$ in $\CC(\A)$ there is an associated localization
endofunctor $\ell$ in $\D(\A)$. This means that $\ell$ is
idempotent (in a natural way) and that the objects that go to 0 by
$\ell$ are those of the smallest localizing (= triangulated and
stable for coproducts) subcategory of $\D(\A)$ that contains $E$.
As applications, we construct K-injective resolutions for complexes
of objects of $\A$ and derive Brown representability for $\D(\A)$
from the known result for $\D(R\text{-}\mathbf{mod})$, where $R$ is
a ring with unit.

It is shown that a morphism of quivers having a certain path
lifting property has a decomposition that mimics the decomposition
of maps of topological spaces into homotopy equivalences composed
with fibrations. Such a decomposition enables one to describe the
right adjoint of the restriction of the representation functor
along a morphism of quivers having this path lifting property.
These right adjoint functors are used to construct injective
representations of quivers. As an application, the injective
representations of the cyclic quivers are classified when the base
ring is left noetherian. In particular, the indecomposable
injective representations are described in terms of the injective
indecomposable $R$-modules and the injective indecomposable
$R[x,x^{-1}]$-modules.

We settle a conjecture of Goresky, Kottwitz and MacPherson related
to Koszul duality, \ie, to the correspondence between differential
graded modules over the exterior algebra and those over the
symmetric algebra.

Most extant localization theories for spaces, spectra and diagrams
of such can be derived from a simple list of axioms which are verified
in broad generality. Several new theories are introduced, including
localizations for simplicial presheaves and presheaves of spectra at
homology theories represented by presheaves of spectra, and a theory
of localization along a geometric topos morphism. The
$f$-localization concept has an analog for simplicial presheaves, and
specializes to the $\hbox{\Bbbvii A}^1$-local theory of
Morel-Voevodsky. This theory answers a question of Soul\'e concerning
integral homology localizations for diagrams of spaces.